Student Number : 0413210T - MSc dissertation - School of Mathematics - Faculty of Science / The aim of this work is to do a literature study on spaces of martingales on Riesz spaces and Banach lattices, using [16, 19, 20, 17, 18, 2, 30] as a point of departure. Convergence of martingales in the classical theory of stochastic processes has many applications in mathematics and related areas. Operator theoretic approaches to the classical theory of stochastic processes and martingale theory in particular, can be found in, for example, [4, 5, 6, 7, 13, 15, 26, 27]. The classical theory of stochastic processes for scalar-valued measurable functions on a probability space ( ,#6;, μ) utilizes the measure space ( ,#6;, μ), the norm structure of the associated Lp(μ)-spaces as well as the order structure of these spaces. Motivated by the existing operator theoretic approaches to classical stochastic processes, a theory of discrete-time stochastic processes has been developed in [16, 19, 20, 17, 18] on Dedekind complete Riesz spaces with weak order units. This approach is measure-free and utilizes only the order structure of the given Riesz space. Martingale convergence in the Riesz space setting is considered in [18]. It was shown there that the spaces of order bounded martingales and order convergent martingales, on a Dedekind complete Riesz space with a weak order unit, coincide. A measure-free approach to martingale theory on Banach lattices with quasi-interior points has been given in [2]. Here, the groundwork was done to generalize the notion of a filtration on a vector-valued Lp-space to the M-tensor product of a Banach space and a Banach lattice (see [1]). In [30], a measure-free approaches to martingale theory on Banach lattices is given. The main results in [30] show that the space of regular norm bounded martingales and the space of norm bounded martingales on a Banach lattice E are Banach lattices in a natural way provided that, for the former, E is an order continuous Banach lattice, and for the latter, E is a KB-space. The definition of a ”martingale” defined on a particular space depends on the type of space under consideration and on the ”filtration,” which is a sequence of operators defined on the space. Throughout this dissertation, we shall consider Riesz spaces, Riesz spaces with order units, Banach spaces, Banach lattices and Banach lattices with quasi-interior points. Our definition of a ”filtration” will, therefore, be determined by the type of space under consideration and will be adapted to suit the case at hand. In Chapter 2, we consider convergent martingale theory on Riesz spaces. This chapter is based on the theory of martingales and their properties on Dedekind complete Riesz spaces with weak order units, as can be found in [19, 20, 17, 18]. The notion of a ”filtration” in this setting is generalized to Riesz spaces. The space of martingales with respect to a given filtration on a Riesz space is introduced and an ordering defined on this space. The spaces of regular, order bounded, order convergent and generated martingales are introduced and properties of these spaces are considered. In particular, we show that the space of regular martingales defined on a Dedekind complete Riesz space is again a Riesz space. This result, in this context, we believe is new. The contents of Chapter 3 is convergent martingale theory on Banach lattices. We consider the spaces of norm bounded, norm convergent and regular norm bounded martingales on Banach lattices. In [30], filtrations (Tn) on the Banach lattice E which satisfy the condition 1[n=1 R(Tn) = E, where R(Tn) denotes the range of the filtration, are considered. We do not make this assumption in our definition of a filtration (Tn) on a Banach lattice. Our definition yields equality (in fact, a Riesz and isometric isomorphism) between the space of norm convergent martingales and 1Sn=1R(Tn). The aforementioned main results in [30] are also considered in this chapter. All the results pertaining to martingales on Banach spaces in subsections 3.1.1, 3.1.2 and 3.1.3 we believe are new. Chapter 4 is based on the theory of martingales on vector-valued Lp-spaces (cf. [4]), on its extension to the M-tensor product of a Banach space and a Banach lattice as introduced by Chaney in [1] (see also [29]) and on [2]. We consider filtrations on tensor products of Banach lattices and Banach spaces as can be found in [2]. We show that if (Sn) is a filtration on a Banach lattice F and (Tn) is a filtration on a Banach space X, then 1[n=1 R(Tn Sn) = 1[n=1 R(Tn) e M 1[n=1 R(Sn). This yields a distributive property for the space of convergent martingales on the M-tensor product of X and F. We consider the continuous dual of the space of martingales and apply our results to characterize dual Banach spaces with the Radon- Nikod´ym property. We use standard notation and terminology as can be found in standard works on Riesz spaces, Banach spaces and vector-valued Lp-spaces (see [4, 23, 29, 31]). However, for the convenience of the reader, notation and terminology used are included in the Appendix at the end of this work. We hope that this will enhance the pace of readability for those familiar with these standard notions.

In this thesis, we review basic properties of linear codes and lattices with a certain focus on their interplay. In particular, we focus on the analogous con- structions of cyclic codes and cyclic lattices. We start out with a brief overview of the basic theory and properties of linear codes. We then demonstrate the construction of cyclic codes and emphasize their importance in error-correcting coding theory. Next we survey properties of lattices, focusing on algorithmic lattice problems, exhibit the construction of cyclic lattices and discuss their applications in cryptography. We emphasize the similarity and common prop- erties of the two cyclic constructions.

Engenharia de ecossistemas refere-se à habilidade de certos organismos de modificar ativamente o ambiente que os cerca. No contexto ecológico, engenheiros de ecossistemas são espécies-chave que modificam ou criam habitats por meios mecânicos ou usando suas próprias estruturas corporais. Ao criarem novos nichos, castores, recifes de corais e sociedades humanas primitivas garantem tanto a própria existência, quanto a de outros organismos no mesmo ecossistema. Devido a seu caráter de longa duração, algumas destas modificações podem persistir até mais do que a duração de uma população de engenheiros, implicando em consequências evolutivas. O estudo teórico de tal fenômeno ecológico é relativamente recente se comparado com a descrição de interações tipo predador-presa ou de competição. Apenas em 1996 Gurney & Lawton introduziram um modelo descrevendo a dinâmica populacional de engenheiros de ecossistemas, mas a partir de lá poucas modificações apareceram. Aqui nós complementamos tal modelo ao permitir que engenheiros se movam difusivamente através dos sítios de um mapa acoplado, uma formulação discreta no espaço e no tempo. A análise de estabilidade local revela a existência dos regimes estável, cíclico e caótico, com uma cascata de bifurcações levando a órbitas caóticas. Obtemos que apenas para altas taxas de crescimento, onde ocorre comportamento caótico, a dispersão influencia na dinâmica das metapopulações. Neste regime, o caos é suprimido e a extinção pode ser evitada. / Ecosystem engineering refers to the ability of certain organisms to actively modify their surrounding environment. In an ecological context, ecosystem engineers are keystone species that modify or create habitats via mechanical means or by using their own physical structures. By creating new niches, beavers, coral reefs and primitive human societies would guarantee both their and other species survival in a shared ecosystem. Due to its long-lasting character, some of this changes might outlive the engineers populations, leading to evolutionary consequences. The theoretical study of such ecological phenomena is relatively recent when compared to the description of predator-prey or competition interactions. Only in 1996 Gurney & Lawton introduced a model to describe the population dynamics of ecosystem engineers, yet since then few modifications appeared. Here we build on this model by allowing the engineers to move diffusively through the patches of a coupled map lattice, a framework discrete both in time and space. The local stability analysis reveals the existence of stable, cyclic and chaotic regimes, with period-doubling bifurcations leading to chaotic orbits. We find that only for large intrinsic growth rates, where chaotic behavior occurs, dispersal influences the metapopulation dynamics. In this regime, chaos is suppressed and extinction can be avoided.

Espectros Raman de super-redes &#948-dopadas, na geometria de retro¬espalhamento, mostram uma estrutura na região do fônon LO que é alargada e deslocada em relação à correspondente linha observada no material intrínseco. Neste trabalho, este fenômeno é teoricamen¬te explicado através do acoplamento de um fônon LO de momento q, propagando-se ao longo da super-rede, com as excitações eletrônicas desse sistema. O deslocamento e o alargamento da linha são obtidos através do cálculo da auto-energia desse fônon, a qual pode ser expressa em termos da função resposta densidade-densidade do gás de elétrons modulado. Efeitos da interação elétron-elétron são tratados através da aproximação das fases aleatórias (RPA). / The Raman spectra of &#948-doping superlattice, in backscattering ge¬ometry, show a structure in the region of the LO phonon which is broadened and shifted in comparison whit the same structure in the intrinsic material. In this work, this phenomena is interpreted as consequence of the coupling of LO xne-phonon of momentum q, along the axis of the superlattice, whit the excitations of the modulate electron gas, that exists in this system. The shift and the broadening of the phonon are calculated as the real and imagi¬nary parts of the phonon self-energy, which in turn is related to the density-density response function of modulate electron gas. Effects of electron-electron interaction are calculated within the Random Phase Approximation (RPA).

Sincronismo é um comportamento dinâmico que pode ser detectado na atividade dos neurônios que formam estruturas corticais, e essa atividade parece ser fundamental para a realização de processos relacionados a aprendizado, compreensão e reconhecimento. Estruturas corticais podem ser modeladas por redes de mapas acoplados. Nessas redes, a existência de solução síncrona é determinada pela equação que governa a atividade de cada mapa, pela topologia de acoplamento entre os mapas e pelos valores dos parâmetros. O enfoque deste trabalho é empregar redes de mapas acoplados voltados à detecção de simetria, e segmentação de imagens, via sincronismo. / Synchronism is a dynamical behavior that can be found in the neural activity of cortical structures, and this behavior seems to be fundamental in learning, understanding and recognition processes. Cortical structures can be modeled by using coupled map lattices. In such lattices, the existence of synchronous solution is determined by the equation that rules the activity of each map, by the coupling topology among the maps, and by the parameter values. The goal of this work is to employ coupled map lattices in symmetry detection and image segmentation, via synchronism.

Reticulados têm sido aplicados de diferentes maneiras em criptografia. Inicialmente utilizados para a destruição de criptossistemas, eles foram posteriormente aplicados na construção de novos esquemas, incluindo criptossistemas assimétricos, esquemas de assinatura cega e os primeiros métodos para encriptação completamente homomórfica. Contudo, seu desempenho ainda é proibitivamente lenta em muitos casos. Neste trabalho, expandimos técnicas originalmente desenvolvidas para encriptação homomórfica, tornando-as mais genéricas e aplicando-as no esquema GGH-YK-M, um esquema de encriptação de chave pública, e no esquema LMSV, a única construção homomórfica que não sucumbiu a ataques de recuperação de chaves IND-CCA1 até o momento. Em nossos testes, reduzimos o tamanho das chaves do GGH-YK-M em uma ordem de complexidade, especificamente, de O(n2 lg n) para O(n lg n), onde n é um parâmetro público do esquema. A nova técnica também atinge processamento mais rápido em todas as operações envolvidas em um criptossistema assimétrico, isto é, geração de chaves, encriptação e decriptação. A melhora mais significativa é na geração de chaves, que se torna mais de 3 ordens de magnitude mais rápida que resultados anteriores, enquanto a encriptação se torna por volta de 2 ordens de magnitude mais rápida. Para decriptação, nossa implementação é dez vezes mais rápida que a literatura. Também mostramos que é possível aumentar a segurança do esquema LMSV contra os ataques quânticos de recuperação de chaves recentemente publicados pela agência britânica GCHQ. Isso é feito através da adoção de reticulados não-ciclotômicos baseados em anéis polinomiais irredutíveis quase-circulantes. Em nossa implementação, o desempenho da encriptação é virtualmente idêntico, e a decriptação torna-se ligeiramente inferior, um pequeno preço a se pagar pelo aumento de segurança. A geração de chaves, porém, é muito mais lenta, devido à necessidade de se utilizar um método mais genérico e caro. A existência de métodos dedicados altamente eficientes para a geração de chaves nesta variante mais segura do LMSV permanece como um problema em aberto. / Lattices have been applied in many different ways in cryptography. Firstly used for the destruction of cryptosystems, they were later applied in the construction of new schemes, including asymmetric cryptosystems, blind signature schemes and the first methods for fully homomorphic encryption. Nonetheless, performance is still prohibitively slow in many cases. In this work, we expand techniques originally devised for homomorphic encryption, making them more general and applying them to the GGH-YK-M cryptosystem, a lattice-based public-key cryptosystem, and to the LMSV scheme, the only known homomorphic scheme that has not succumbed to INDCCA1 key recovery attacks to this date. In our tests, we reduce public key bandwidth occupation of GGH-YK-M by an order of complexity, specifically, from O(n2 lg n) down to O(n lg n) bits, where n is a public parameter of the scheme. The new technique also attains faster processing in all operations involved in an asymmetric cryptosystem, that is, key generation, encryption, and decryption. The most significant improvement in performance is in key generation, which becomes more than 3 orders of magnitude faster than previous results, while encryption becomes about 2 orders of magnitude faster. For decryption, our implementation is ten times faster than the literature. We also show that it is possible to improve security of LMSV against the quantum key recovery attacks recently published by British GCHQ.We do so by adopting non-cyclotomic lattices based on nearly-circulant irreducible polynomial rings. In our implementation, performance of encryption remains virtually the same, and decryption becomes slightly worse, a small price to pay for the improved security. Key generation, however, is much slower, due to the fact that it is necessary to use a more generic and expensive method. The existence of highly effcient dedicated methods for key generation of this secure variant of LMSV remains as an open problem.

Thesis (Elec.E)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Bibliography: leaves 172-176. / by Chon Meng Wong. / Elec.E

For the study of cobalt oxide materials, we investigate the spin susceptibility behavior when the temperature and interaction strength vary in the whole filling region, on a triangular lattice. We find the high density of state (DOS) around the van Hove singularity point is important to the existence of a certain itinerant ferromagnetic short-range correlation. Also we study different spin singlet and triplet pairing channels in this region, and find the triplet channels especially the f-wave pairing susceptibility presents the quickest growth via the lowing of temperature. However, it is subtle to stabilize the f-wave pairing since the effective pairing interaction was found to be very small. / In the last part we discuss a special case in quantum phase transition, in which the Green's function of the system can be analytically obtained. We study the entanglement of the system to identify the quantum phase transition in the system. / We investigate several novel strong correlated phenomena in the geometrically frustrated system in this thesis. On a parallel computational grid, we do the state-of-the-art quantum Monte Carlo (QMC) numerical simulation to the extended single-band Hubbard model. We observe a wide range of measurements to clarify and demonstrate the system properties in different electron density, temperature, interaction strength. The results of the single-band Hubbard model on the triangular lattice can help us understand the magnetic property and superconductivity of the cobalt oxide materials. The results of the attractive Hubbard model on the honeycomb lattice can be used to discuss some properties of the "Bardeen-Cooper-Schrieffer (BCS) state to Bose-Einstein condensation (BEC) crossover" picture, such as Dirac Fermions' behaviors and the coexistence of fermionic and bosonic degrees of freedom in the system. / We study the BCS-BEC picture of high Tc superconductor systems. On a honeycomb lattice, we measure a wide range of observables, such as double occupancy, uniform spin susceptibility, on site pair correlation vertex, on-site pair correlation length, potential energy, kinetic energy, to monitor the BEC-type and BCS-type behaviors coexisting in the system. In different regions of the interaction strength-temperature parametric plane, we find that the BEC-type and BCS-type behaviors dominate at different regions of the interaction strength-temperature parametric plane, and they have crossover in the intermediate interaction strength region. / Su, Shiquan. / Adviser: Hai-Qing Lin. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3570. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 101-107). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307. / Su, Shiquan.

Submitted by NAÍSA CAMILA GARCIA null (naisacamila@hotmail.com) on 2017-02-23T13:24:13Z No. of bitstreams: 1 Dissertação de Mestrado Naísa.pdf: 926270 bytes, checksum: e0ef770d876850618bb4fff10a0da639 (MD5) / Approved for entry into archive by LUIZA DE MENEZES ROMANETTO (luizamenezes@reitoria.unesp.br) on 2017-03-02T14:21:45Z (GMT) No. of bitstreams: 1 tosti_ncg_me_sjrp.pdf: 926270 bytes, checksum: e0ef770d876850618bb4fff10a0da639 (MD5) / Made available in DSpace on 2017-03-02T14:21:45Z (GMT). No. of bitstreams: 1 tosti_ncg_me_sjrp.pdf: 926270 bytes, checksum: e0ef770d876850618bb4fff10a0da639 (MD5) Previous issue date: 2017-02-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O propósito deste trabalho é estudar alguns corpos abelianos, mais especificamente, as extensões reais maximais contidas nos corpos ciclotômicos de grau 8 e, os subcorpos dos corpos ciclotômicos Q(ζ_7) e Q(ζ_17). Em tais corpos, determinamos base integral, discriminante, grupo de Galois e construimos submódulos de posto máximo do anel dos inteiros algébricos com sua respectiva representação geométrica. Além disso, calculamos a densidade de centro destes reticulados. / The purpose of this work is to investigate some Abelian Number Fields, especifically the maximal extension contained in the cyclotomic fields of degree 8, and the subfields of the cyclotomic fields Q(ζ_7) and Q(ζ_17). In such fields, we compute: integral bases, discriminant, Galois group and submoduli with maximal rank in the ring of algebraic integers, its geometrical realization with the respective center density.

Orientador: Antonio Aparecido de Andrade / Banca: Henrique Lazari / Banca: Tatiana Betoldi Carlos / Resumo: O objetivo deste trabalho é o estudo de reticulados ideais. Neste estudo enfatizamos o artigo "Lattices and Number Fields" de Eva Bayer-Fluckiger, que apresenta alguns reticulados ideais com as mesmas propriedades que os reticulados Ap¡1, p primo, D4, E6, E8, K12 e ¤24. / Abstract: The aim of this work is the study of ideal lattices. In this study we stress a Eva Bayer-Fluckiger's article \Lattices and Number Fields" with presents some ideal lattices with same properties that lattices Ap¡1, p prime number, D4, E6, E8, K12 e ¤24. / Mestre